Feedback Linearizability of Strict Feedforward Systems

Transcription

1 Proceedings o the 47th IEEE Conerence on Decision and Control Cancun, Mexico, Dec 9-11, 28 WeB11 Feedback Linearizability o Strict Feedorward Systems Witold Respondek and Issa Amadou Tall Abstract For any strict eedorward system that is eedback linearizable we provide (ollowing our earlier results an algorithm, along with explicit transormations, that linearizes the system by change o coordinates and eedback in two steps: irst, we bring the system to a newly introduced Nonlinear Brunovský canonical orm (NBr and then we go rom (NBr to a linear system The whole linearization procedure includes dieo-quadratures (dierentiating, integrating, and composing unctions but not solving PDE s Application to eedback stabilization o strict eedorward systems is given I INTRODUCTION Consider a smooth nonlinear single-input control system Ξ : ż = F (z, u, z Z R n, u R, where z Z, an open subset o R n Assume that it is, via a smooth change o coordinates x = ϕ 1 (z, equivalent to the strict eedorward orm, shortly (SF F -orm, (SF F ẋ 1 = G 1 (x 2,, x n, u ẋ = G (x n, u ẋ n = G n (u I the system Ξ is eedback linearizable, then (as it is well known, see, eg [2], [4], [12] it takes, in some coordinate system y = ϕ 2 (x, the eedback orm, shortly (F B-orm: (F B ẏ 1 = Ḡ1(y 1, y 2 ẏ = Ḡ(y 1,, y n ẏ n = Ḡn(y 1,, y n, u I Ξ takes in some x-coordinates the (SF F -orm and in some y-coordinates the (F B-orm, then a natural question arises whether there exist coordinates w = ϕ 3 (z in which Ξ would take simultaneously both the (SSF -orm and the (F B-orm Comparing (SF F and (F B, that are dual with respect to each other, we see that in w-coordinates (i they exist, Ξ would take the ollowing nonlinear generalization o the Brunovský canonical orm: (NBr ẇ 1 = Ĝ1(w 2 ẇ = Ĝ(w n ẇ n = Ĝn(u Recall that the Brunovský canonical orm is the ollowing linear control system on R n (consisting o a chain o n W Respondek is with INSA-Rouen, Laboratoire de Mathématique, Mont Saint Aignan, France wresp@insa-rouenr IA Tall is with the Department o Mathematics, Southern Illinois University, Carbondale IL 6291, USA itall@mathsiuedu integrators: (Br w 1 = w 2 w = w n w n = ũ One o the main results o this paper asserts that the answer to the above question is indeed positive: i a system is via (dierent, in general changes o coordinates equivalent to the (SF F -orm and to the (F B-orm, then it is also equivalent to the above Nonlinear Brunovský canonical orm (NBr, which is simultaneously (SF F and (F B A similar question arises i we assume that Ξ is equivalent to the (SF F -orm and, instead o supposing eedback linearizability, we assume that it is linearizable via a change o coordinates only Then we would like to know whether the system can be put into a linear orm that would simultaneously be (SF F This question was answered positively in our previous paper [19], inspired by that o Krstic [7], where we provided an algorithm, along with necessary and suicient conditions, to linearize (via a change o state coordinates a strict eedorward system Let us recall that strict eedorward systems and their stabilization were irst investigated by Teel in his pioneering papers [2], [21] Since then, it has been ollowed by a growing literature [1], [5], [6], [7], [8], [9], [1], [11], [13], [14], [15], [17], [18] Recently, Krstic [7] addressed the problem o linearizability o nonlinear systems in strict eedorward orm, and provided two classes (type I and type II that are linearizable by change o coordinates By providing linearizing changes o coordinates in some examples, Krstic mentioned the lack o a systematic way o inding those changes o coordinates We addressed this problem in [19] and provided an eicient algorithm to ind linearizing transormations or strict eedorward systems that are linearizable by change o state coordinates The aim o this paper is to study the class o eedback linearizable strict eedorward systems in its ull generality The problem o transorming a control system into a linear controllable system via change o coordinates and eedback was solved in the early eighties in [2] and [4], where necessary and suicient geometric conditions, expressed in terms o involutivity o certain distributions (see Theorem IV4 below, were obtained (see also [3], [12] Those conditions are easy to check but i they are satisied, then inding linearizing coordinates and eedback transormations requires, in general, solving a system o partial dierential equations (whose solvability is guaranteed by the involutivity For strict eedorward systems, however, /8/$25 28 IEEE 2499

2 47th IEEE CDC, Cancun, Mexico, Dec 9-11, 28 WeB11 inding linearizing coordinates and eedback turns out to be much easier: our algorithm can be perormed using at most n( 2 steps, quadratures, each involving composition and integration o unctions only (but not solving PDEs ollowed by a sequence o n derivations That simplest possible way o calculating linearizing eedback transormations (using dieo-quadratures only which is crucial or applications or any strict eedorward system shows importance o the presented algorithm Moreover, i the system is not eedback linearizable, the algorithm ails ater a inite number o steps, which thus provides a simple way o testing the eedback linearizability o strict eedorward systems The paper is organized as ollows In Section II we give some basic notations In Section III we ormulate our main result on F -linearizable strict eedorward systems Then we show that the constructed transormations are, indeed, dieoquadratures in Section IV and discuss applications to the stabilization problem in Section V Finally, the proo o the main result, together with the Algorithm, orm Section VI II DEFINITIONS AND NOTATIONS Throughout the paper, the word smooth will always mean C -smooth We assume, except otherwise stated, that Ξ is aine in control, ie, 2 F u =, and we denote it by Σ 2 Consider two control systems Σ : ż = (z + g(zu, z Z R n, u R, where and g are smooth vector ields on Z, an open subset o R n, and Σ : z = ( z + g( zũ, z Z R n, ũ R, where and g are smooth vector ields on Z, an open subset o R n They are called state equivalent, shortly S-equivalent, i there exists a smooth dieomorphism ϕ : Z Z, such that ϕ = and ϕ g = g; (we take u = ũ Recall that or any smooth vector ield on Z and any smooth dieomorphism z = ϕ(z we denote (ϕ ( z = Dϕ(z (z, with z = ϕ 1 ( z Two control systems Σ and Σ are called eedback equivalent, shortly F-equivalent, i there exist a smooth dieomorphism ϕ : Z Z and smooth R-valued unctions α, β, satisying β(, such that ϕ ( + gα = and ϕ (gβ = g A control-aine system that is strict eedorward takes the ollowing aine strict eedorward orm ż 1 = 1 (z 2,, z n + g 1 (z 2,, z n u (ASF F ż = (z n + g (z n u ż n = n + g n u, where n, g n R and z Z, an open subset o R n Throughout the paper we assume that the drit has an equilibrium which, without loss o generality, is taken to be R n, that is ( = In particular, n = in (ASF F III MAIN RESULT: F-LINEARIZABLE (SFF-SYTEMS In this section we will give our main result on F - linearizable strict eedorward systems Theorem III1 Assume that a control-aine system Σ is S- equivalent to the (ASF F -orm Then the ollowing conditions are equivalent: (i Σ is F -equivalent to a linear controllable system; (ii Σ is S-equivalent to the ollowing Aine Nonlinear (AN Br Brunovský orm: ẇ 1 = Ĝ 1 (w 2 ẇ = Ĝ (w n ẇ n = u Remark III2 The above theorem holds both locally and globally More precisely, assume that Σ is, locally at Z R n, S-equivalent to the (ASF F -orm Then (i and (ii are equivalent locally around z Now assume that Σ is globally S-equivalent to the (ASF F -orm on R n, then (i, satisied locally around any point z Z, is equivalent to the global S-equivalence to (ANBr on R n Proo o this Theorem, together with the Algorithm on which it is based, is given in Section VI O course, i Σ is F -linearizable (that is, i (i o the above theorem holds, then it is F -equivalent to the Brunovský (Br-orm Our result states, that F -linearizable systems that are S-equivalent to the aine strict eedorward (ASF F -orm exhibit also a nice orm under a change o coordinates only Namely, they are S-equivalent to the aine nonlinear Brunovský (ANBr- orm I we consider a general nonlinear system Ξ and assume that it is S-equivalent (locally or globally to the (SF F - orm, then the above theorem remains valid (locally or globally with the orm (ANBr replaced by (NBr, that is, the equation ẇ n = u in (ii replaced by ẇ n = Ĝn(u IV CALCULATING NORMALIZING AND LINEARIZING TRANSFORMATIONS In this section we will examine the class o state and eedback transormations that bring (ASF F -systems to the (AN Br-orm and to the linear Brunovský orm (Br Let us start with the ollowing result o the authors [19], where we studied strict eedorward systems that are S-linearizable Theorem IV1 Assume that Σ is S-equivalent to the (ASF F -orm Then the ollowing conditions are equivalent: (i Σ is S-equivalent to a linear controllable system; (ii Σ is S-equivalent to the Brunovský orm (Br This result was proved (in a slightly dierent context in [19] in a constructive way that gives the linearizing dieomorphism in an explicit orm calculated via integrations and compositions o unctions only We will ormalize this important property as ollows 25

3 47th IEEE CDC, Cancun, Mexico, Dec 9-11, 28 WeB11 We will say that a transormation is calculated by quadratures i it is deined by a inite sequence consisting o elementary operations, composing unctions and calculating integrals We say that a transormation is calculated by dieoquadratures i it is deined by a inite sequence consisting o elementary operations, composing unctions, calculating integrals, and dierentiating We say that two systems are S-equivalent by biquadratures i there exists a dieomorphism ϕ conjugating them so that ϕ and ϕ 1 are calculated by quadratures Proposition IV2 Consider a control-aine system Σ in the aine strict eedorward (ASF F -orm I the system is S-linearizable, then it is S-equivalent to the Brunovský canonical orm (Br by bi-quadratures dieomorphism and the inverse o control transormation involves compositions, integrations, and inverting nonlinear unctions A ew comments are to be given Consider a (controlaine, or simplicity single-input nonlinear system ż = (z + g(zu, which is not necessarily in the (ASSF -orm We attach to Σ the sequence o nested distributions D 1 D 2 D n : { } D k = span g, ad g,, ad k 1 g, k = 1, 2,, n, with ad g = g, and inductively, adk 1 g = [, ad k 2 g] Necessary and suicient conditions or eedback linearization obtained in [2], [4] (see also [3] and [12] are as ollows: For systems in aine strict eedorward (ASF F -system that are F -linearizable, described by Theorem III1, the picture is slightly dierent Proposition IV3 Consider a control system Σ in the aine strict eedorward (ASF F -orm I the system is F - linearizable, then it is S-equivalent to the aine nonlinear Brunovský orm (AN Br by bi-quadratures Moreover, it is F -transormable to the Brunovský canonical orm (Br (and thus F -linearizable by a transormation that is calculated by dieo-quadratures Proo The proo o the irst statement ollows directly rom the Algorithm given in Section VI which provides explicit ormulas or the components o the dieomorphism w = ϕ(z transorming (ASF F into (AN Br Those ormulas involve, indeed, compositions, elementary operations, and integrations only Notice that at the irst glance we can suspect that in order to ind ϕ l (z (see the general substep o the general step o Algorithm we have to integrate b l and in order to know b l we have to calculate derivatives since b l = l ( i 1 zi 1 It is crucial to observe that there exists another way to calculate b l as b l = ( i 1 (z i 1 ( l (z l+1,, z i (z l+1,, z i 1, which involves composition and elementary operations only Now, consider the (AN Br-orm and denote (w = Ĝ1(w 2 ++Ĝ(w n, g(w = w 1 w w n It is well known (see, eg, [3] and [12] that in order to transorm (via eedback (AN Br into (Br we use the ollowing inite sequence o derivations (LF w 1 = h(w, w i = L i 1 h(w, i = 2,, n, ũ = L n h(w + (L gl h(wu, where h(w = w 1 Notice that we do not claim that Σ is F -equivalent to a linear system (to the Brunovský orm (Br, or instance by bidieo-quadratures Indeed, calculating the inverse eedback transormation (actually both, the inverse o the linearizing 251 Theorem IV4 A control-aine system Σ : ż = (z+g(zu is locally equivalent, via a change o coordinates w = ϕ(z and eedback ũ = α(z + β(zu, to a linear controllable Brunovský canonical orm (Br i and only i (F1 dim D n (z = n (F2 D is involutive As it is well known (see, eg, [2], [3], [4], [12], in order to F -linearize Σ we have to ind a linearizing output, that is a unction h whose dierential dh does not vanish and annihilates the involutive distribution D Then h deines the linearizing coordinates and linearizing eedback by the ormula (LF given in the proo o Proposition IV3 In order to ind h we have to solve the system o irst order PDE s L g L i 1 h =, 1 i n 1, L g L h This system admits a solution (assured by involutivity o D but its solvability is, in general, a highly nontrivial task A partial corollary o Proposition IV3 is that or F - linearizable systems Σ that are in the (ASF F -orm, the problem o inding a linearizing output h can by solved by quadratures Indeed, the irst statement o Proposition IV3 asserts that we can ind, by quadratures (o the components i (z and g i (z, the dieomorphism w = ϕ(z that transorms Σ into the Aine Nonlinear Brunovsky canonical orm (AN Br A linearizing output is now the irst component w 1 = ϕ 1 (z o that dieomorphism (compare the second part o the proo o Proposition IV3 to see this V STABILIZATION OF F-LINEARIZABLE (ASFF-SYSTEMS It is well known that any F -linearizable system is (locally asymptotically stabilizable by a state eedback that is linear with respect to the linearizing coordinates, see, eg, [3] The diiculty o implementing this result resides on the act that the linearizing coordinates and eedback law are not always easy to ind For F -linearizable system that are in the (ASF F -orm, our algorithm provides, however, an easy way o inding the linearizing transormations and, as a consequence, a stabilizing controller via dieo-quadratures Namely, Proposition IV3 implies the ollowing result:

4 47th IEEE CDC, Cancun, Mexico, Dec 9-11, 28 WeB11 Proposition V1 Consider a system Σ : ż = (z+g(zu in (ASF F -orm, locally around R n (resp globally on R n that is F -linearizable locally at R n (resp locally at any z R n Let w = ϕ(z be the coordinates change (given by the Algorithm that takes Σ into the (ANBr-orm, and w = ψ(w, ũ = α(w+ β(wu, given by (LF in the proo o Proposition IV3, be the eedback transormation that maps the (AN Br-orm into the (Br-orm Then the eedback law α(ϕ(z + n k i ψ i (ϕ(z u = β(ϕ(z, (1 where the polynomial p(λ = λ n + i= λi k i+1 is Hurwitz, locally (resp globally on R n asymptotically stabilizes the origin R n Moreover, this stabilizing control law can be calculated by dieo-quadratures (in terms o the components i (z and g i (z o the original system Σ Proo The proo is a direct consequence o Proposition IV3 and stabilizability o eedback linearizable systems Indeed, applying the controller (1 yields the closed loop linear system w = A w, where w = Φ(z = ψ ϕ(z, and A is Hurwitz Thus, by Proposition IV3, the components Φ i o Φ are calculated by dieo-quadratures in terms o the components i (z and g i (z o the original system Σ It is interesting to observe that the Lyapunov unction V can also be calculated by dieo-quadratures (in terms o the components o the original system Indeed, let P be the positive deinite symmetric matrix solution o the Riccati equation A P + P A = I Then V (z = Φ (zp Φ(z To illustrate this result we consider the ollowing example Example V2 Let us consider the system (ASF F ż 1 = sin(z 2 + z 2(z 2 + z sin z 3, ż 2 = sin z 3 sin z n, ż i = sin z i+1, 3 i n 1 ż n = u, ( which is control-normalized, ie, g(z =,,, 1 The change o coordinates (see Example V2-bis below w = ϕ(z w 1 = z 1 + (z 2 + z 2, w 2 = z 2 + z, w i = z i, 3 i n transorms the system into the (AN Br-orm { ẇi = sin w i+1, 1 i n 1 ẇ n = u, where (w 1,, w n ( π, π ( π, π It is thus eedback linearizable by w = ψ(w, ũ = α(w + β(wu : ψ 1 ψ ψ 1 = w 1, ψ 2 = sin w 2,, ψ n = sin w i+1, w 1 w i ψ n ũ = sin w i+1 + ψ n u w i w n Since ψ i w i = cos w i ψ i 1 w i 1, i = 2,, n, then ψ n w n = cos w n ψ w = = cos w n cos w cos w 2, and thus or any < ɛ < π/4, the eedback law ψ n n sin z i+1 (ϕ(z + k i ψ i (ϕ(z z i u = cos(z 2 + z cos z 3 cos z n locally asymptotically stabilizes the system on ( ɛ, ɛ n VI PROOF OF THEOREM III1 (i (ii It is clear that a system in (ANBr-orm is F - linearizable by the change o coordinates and eedback (LF (ii (i We show that an F -linearizable (ASF F -orm, can be taken, via a (local dieomorphism, to the (ANBr-orm Algorithm Assume that Σ is control-normalized (see [19] It is well known that the involutivity o D (see (F 2 o Theorem IV4 implies that o all distributions D k, 1 k n Step 1 The involutivity o D 2 implies [g, ad g] = γ 1 ad g + γ g, where γ 1 and γ are smooth unctions Because (z = j (z j+1,, z n zj and g = zn, it ollows that γ = and γ 1 = γ 1 (z n Above, zj = z j Thus the involutivity o D 2 reduces to the condition (F n 2 j z 2 n = γ 1 (z n j z n or all 1 j n 1 Condition (F n is necessary or F -linearization, ie, i it ails (γ 1 depends on other variables than z n or γ 1 is not the same or all components j then the algorithm stops I (F n holds, we can simpliy the system using n 2 substeps Let j = n 1 in (F n Since = (z n, we get (z n = γ 1 (z n (z n, which gives γ 1 uniquely as γ 1 = (z n / (z n Two successive integrations yield zn ( t (z = a exp γ 1 (sds dt, with a R = R \ Substep 1 Take j = n 2 in (F n and denote h n 2 = n 2 z n We obtain ater integration ( zn h n 2 (z, z n = a n 2 (z exp γ 1 (sds which implies, ater a second integration, that n 2 (z, z n = c n 2 (z + (z n b n 2 (z, or some smooth unctions c n 2 and b n 2 = a n 2 /a The dieomorphism x = ϕ(z whose components are z j = ϕ j (z = z j, j n 2 z n 2 = ϕ n 2 (z = z n 2 z b n 2 (sds 252

5 47th IEEE CDC, Cancun, Mexico, Dec 9-11, 28 WeB11 transorms the system, by quadratures, into the orm Σ : z = ( z + g( zu, z R n, with g( z = (,,, 1 and ( z = n 3 j ( z j+1,, z n zj + n 2 ( z zn 2 + ( z n z General Substep Assume that or some 1 i n 2, a sequence o quadratures exists whose composition has brought the original system into (we keep the z-notation Σ : ż = (z + g(zu, z R n, with g(z = (,,, 1 and (z = i j (z j+1,, z n zj + n 2 j=i+1 j (z j+1,, z zj + (z n z Taking j = i in the condition (F n we have The original system is thus brought, via n 2 substeps, to Σ : ż = (z + g(zu, z R n, with g = (,,, 1 and (z = i j (z j+1,, z n zj + n 2 j=i+1 j (z j+1,, z zj + (z n z This ends the irst step o the algorithm We will denote by ϕ 1 the composition o the dieomorphisms o step 1 General Step For simplicity, we skip the tildes Assume that Σ has been brought, via quadratures, to the orm Σ : ż = (z + g(zu, z R n, where g = (,,, 1 and or some 3 i n 2 i 2 (z = j (z j+1,, z i zj + j=i 1 j (z j+1 zj We will show that Σ can be brought, via quadratures, to Σ : ( z + g( zu, where g = (,,, 1 and 2 i z 2 n = γ 1 (z n i z n Denoting h i (z i+1,, z n = i zn, we obtain ( zn h i = a i (z i+1,, z exp γ 1 (sds and ater a second integration i = c i (z i+1,, z + (z n b i (z i+1,, z, or some smooth unctions c i and b i = a i /a The dieomorphism x = ϕ(z whose components are z j = ϕ j (z = z j, j i z i = ϕ i (z = z i z b i (z i+1,, z n 2, sds transorms the system, by quadratures, into the orm Σ : z = ( z + g( zu, z R n, with g( z = (,,, 1 and ( z = i 1 + n 2 j=i j ( z j+1,, z n zj j ( z j+1,, z zj + ( z n z Notice that, at each substep, the inverse ψ o the dieomorphism x = ϕ(z is easily computable as z j = ψ j ( z = z j, z i = ψ i ( z = z i + j i z b i ( z i+1,, z n 2, sds Moreover, or any 1 j n 2, we have j ( z j+1,, z = j (ψ j+1 ( z j+1,, z,, ψ ( z 253 ( z = i 3 j ( z j+1,, z i 1 zj + j=i 2 j ( z j+1 zj We deduce rom above that or any 1 i k n 1 ad n k g = µ k (z k+1,, z n zk + ϑ k (z, where the vector ield ϑ k D n k = span { zk+1,, zn } and µ k is a smooth unction In particular or k = i we have ad n i g = µ i (z i+1,, z n zi + ϑ i (z, rom which, and the expression o, we deduce that ad n i+1 g = i 1 µ j (z j+1,, z n zj + ϑ i 1 (z, where ϑ i 1 n i+1 and or any 1 j i 1 µ j (z j+1,, z n = µ i (z i+1,, z n j A simple calculation shows that [ ad n i+1 ] g, ad n i g i 1 = µ 2 i 2 j zi 2 zj + ϑ i 1 (z, where ϑ i 1 D n i+1 = span { zi,, zn } The involutivity o D n i+2 implies that [ ] n ad n i+1 g, ad n i g = γ n k ad n k g k=i 1 = γ n i+1 ad n i+1 or some smooth unctions γ, γ 1,, γ n i+1 Comparing the two Lie brackets it ollows that i 1 (µ i 2 2 j z 2 i i 1 zj = (µ i γ n i+1 g + ˆϑ i 1 j zj,

6 47th IEEE CDC, Cancun, Mexico, Dec 9-11, 28 WeB11 that is, the condition (F i 2 j z 2 i = γ n i+1 j, 1 j i 1 For j = i 1 we get i 1 (z i = γ n i+1 (z i 1 (z i which determines γ n i+1 = i 1 (z i/ i 1 (z i uniquely in terms o z i The components i 1 and γ n i+1 are related by zi ( t i 1 (z i = a i 1 exp γ n i+1 (sds dt, a i 1 R General Substep Let us assume that the original system has been brought, via quadratures, to the orm Σ : ż = (z + g(zu, z R n, where g = (,,, 1 and or some 2 l < i n 2 (z = l j (z j+1,, z i zj + j=l+1 j (z j+1 zj Taking j = l in (F i and denoting h l = l zi we have ( zi h l = a l (z l+1,, z i 1 exp γ n i+1 (sds and ater integration l (z l+1,, z i = c l (z l+1,, z i 1 + i 1 (z i b l (z l+1,, z i 1, or some smooth unctions c l and b l = a l /a i 1 The new coordinates x = ϕ(z whose components are z j = ϕ j (z = z j, j l z l = ϕ l (z = z l zi 1 b l (z l+1,, z i 2, sds transorms the system, by quadratures, into the orm Σ : z = ( z + g( zu, z R n, where g = (,,, 1 and ( z = l 1 j ( z j+1,, z i zj + j=l j ( z j+1 zj This ends the general step Denote by ϕ i the composition o the coordinates changes or the i-th step Thus the composition ϕ n 2 ϕ 1 deines the coordinates change taking Σ into the (AN Br-orm, which completes the proo o Theorem III1 Example V2-bis Reconsider Example V2 Then (F n holds with γ 1 = tan z n For 3 i n 1, the decomposition i (z = c i (z i+1,, z + (z n b i (z i+1,, z yields b i = and c i = sin z i+1 because i (z = sin z i+1 Moreover, b 2 = 1 and c 2 = sin z 3 since (z = sin z n and 2 (z = sin z 3 sin z n The transormation z j = z j, j 2 z 2 = z 2 z ( 1ds = z 2 + z brings the system into the orm z 1 = sin z 2 2 z 2 sin z 3, z i = sin z i+1, 2 i n 1 z n = u Next, we apply the last step (Step n 3 with (F 3 2 j z 2 3 = γ n 2 ( z j z 3, 1 j 2, which holds or γ n 2 = tan z 3 The decomposition o 1 ( z = c 1 ( z ( z 3 b 1 ( z 2 yields c 1 ( z 2 = sin z 2 and b 1 ( z 2 = 2 z 2 Hence w j = z j, j 1 w 1 = z 1 z2 ( 2sds = z 2 + z 2 2 The composition gives the linearizing coordinates system REFERENCES [1] A Astoli and G Kaliora, A geometric characterization o eedorward orms, in IEEE Trans Autom Contr, 5[7], pp [2] L R Hunt and R Su, Linear equivalents o nonlinear time varying systems, in: Proc MTNS, Santa Monica, CA, (1981, pp [3] A Isidori, Nonlinear Control Systems, 3rd ed, Springer, London, 1995 [4] B Jakubczyk and W Respondek, On linearization o control systems, Bull Acad Polon Sci Ser Math, 28, (198 pp [5] M Jankovic et al, Constructive Lyapunov stabilization o nonlinear cascade systems, IEEE Trans Autom Control, (1996, pp [6] M Jankovic, R Sepulchre, and P Kokotovic, Global Adaptative Stabilization o Cascade Nonlinear Systems, Autom, 32, (1997 pp [7] M Krstic, Feedback linearizability and explicit integrator orwarding controllers or classes o eedorward systems, IEEE Trans Autom Control, 49, (24 pp [8] M Krstic, Explicit Forwarding Controllers-Beyond Linearizable Class, Proceedings o the 25 American Control Conerence, pp [9] A Marigo, Constructive necessary and suicient conditions or strict triangularizability o dritless nonholonomic systems, in Proc 34 th CDC, Phoenix, Arizona, USA, (1999, pp [1] F Mazenc and L Praly, Adding integrations, saturated controls, and stabilization or eedorward orms, IEEE Trans Autom Cont, (1996, pp [11] F Mazenc and L Praly, Asymptotic tracking o a reerence state or systems with a eedorward structure, Automat, 36 (2, pp [12] H Nijmeijer and A J van der Schat, Nonlinear Dynamical Control Systems Springer-Verlag, New York, 199 [13] W Respondek and IA Tall, Smooth and analytic normal and canonical orms or strict eedorward systems, Proc o the 44 th IEEE Con on Decision and Control, Seville, Spain, 25, pp [14] R Sepulchre, M Janković, and P Kokotović Constructive Nonlinear Control, Springer, Berlin-Heidelberg-New York, 1996 [15] R Sepulchre et al, Integrator Forwarding: A New Recursive Nonlinear Robust Design, Automatica, 33, (1997 pp [16] W Respondek and IA Tall, Strict eedorward orm and symmetries o nonlinear control systems, in Proc o the 43 rd IEEE Con on Decision and Control, Atlantis, Bahamas, pp [17] IA Tall and W Respondek, Transorming a single-input nonlinear system to a eedorward orm via eedback, in Nonlinear Control in the Year 2, vol2, A Isidori, F Lamnabhi-Lagarrigue and W Respondek (eds, LNCS vol 259, Springer, London, (2, pp [18] IA Tall and W Respondek, Feedback equivalence to eedorward orms or nonlinar single-input control systems, in Dynamics, Biurctions and Control, F Colonius and L Grüne (eds, LNCS vol 273, Springer, Berlin-Heidelberg, 22, pp [19] IA Tall and W Respondek, On Linearizability o Strict Feedorward Control Systems, in 28 American Control Conerence, pp [2] A Teel, Feedback stabilization: nonlinear solutions to inherently nonlinear problems, Memorandum UCB/ERL M92/65 [21] A Teel, A nonlinear small gain theorem or the analysis o control systems with saturation, IEEE Trans Autom Control, 41 (1996, pp